Problem: Simplify; express your answer in exponential form. Assume $k\neq 0, a\neq 0$. $\dfrac{{(k^{-5}a^{-3})^{-3}}}{{ka^{-2}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(k^{-5}a^{-3})^{-3} = (k^{-5})^{-3}(a^{-3})^{-3}}$ On the left, we have ${k^{-5}}$ to the exponent ${-3}$ . Now ${-5 \times -3 = 15}$ , so ${(k^{-5})^{-3} = k^{15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(k^{-5}a^{-3})^{-3}}}{{ka^{-2}}} = \dfrac{{k^{15}a^{9}}}{{ka^{-2}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{15}a^{9}}}{{ka^{-2}}} = \dfrac{{k^{15}}}{{k}} \cdot \dfrac{{a^{9}}}{{a^{-2}}} = k^{{15} - {1}} \cdot a^{{9} - {(-2)}} = k^{14}a^{11}$